The use of Powell-Sabin B-Splines in a higher-order phase-field model for crack kinking
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ORIGINAL PAPER
The use of Powell-Sabin B-Splines in a higher-order phase-field model for crack kinking Lin Chen1 · Bin Li2 · René de Borst1 Received: 26 April 2020 / Accepted: 1 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Phase-field models for brittle fracture in anisotropic materials result in a fourth-order partial differential equation for the damage evolution. This necessitates a C 1 continuity of the basis functions. Here, Powell-Sabin B-splines, which are based on triangles, are used for the approximation of the field variables as well as for the the description of the geometry. The use of triangles makes adaptive mesh refinement and discrete crack insertion straightforward. Bézier extraction is used to cast the B-splines in a standard finite element format. A procedure to impose Dirichlet boundary condition is provided for these elements. The versatility and accuracy of the approach are assessed in two case studies, featuring crack kinking and zig-zag crack propagation. It is also shown that the adaptive refinement well captures the evolution of the phase field. Keywords Phase-field model · Powell-Sabin B-splines · Anisotropy · Bézier extraction · Adaptive refinement
1 Introduction The analysis of crack propagation remains a challenging problem which comprises initiation, (unstable) propagation, branching, crack interaction, coalescence and merging. A host of numerical models, which include discrete crack models, phase-field models, and particle methods, have been proposed to model these phenomena, e.g. [1–6]. The phase-field modelling of crack propagation starts from the pioneering works of Francfort and Marigo [7], in which a variational, discontinuity-free formulation has been introduced for brittle fracture. The method relies on a regularised description of the discontinuities [8]. In the regularised model, cracks are represented by a scalar phase-field variable c, which varies smoothly in a band of finite width This research has been supported by the European Research Council under Advanced Grant 664734.
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Lin Chen [email protected] Bin Li [email protected] René de Borst [email protected]
1
University of Sheffield, Sheffield S1 3JD, UK
2
Guangdong Technion–Israel Institute of Technology, Shantou, Guangdong 515063, China
from 1 for the completely broken material to 0 away from the crack, and thus provides a damage-like description of the crack [9]. For materials with an isotropic surface energy phase-field models have been shown to predict the crack path fairly accurately [10]. Materials with an anisotropic surface energy also exist, either because of their inherent microstructure, or as a consequence of the manufacturing process. Such an anisotropy can strongly influence the crack path, for instance in single crystals, in geological materials, or in extruded polymers. Experimental results clearly show a different crack propagation behaviour from that in materials with an isotropic surface energy [11,12]. Such materials are characterised
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